Exam 14: Introduction to Time Series Regression and Forecasting

Find data for real GDP (Yt)for the United States for the time period 1959:I (first quarter)to 1995:IV.Next generate two growth rates: The (annualized)quarterly growth rate of real GDP [(lnYt - lnYt-1)× 400] and the annual growth rate of real GDP [(lnYt - lnYt-4)× 100].Which is more volatile? What is the reason for this? Explain.

The forecast is

The Bayes-Schwarz Information Criterion (BIC)is given by the following formula

The formulae for the AIC and the BIC are different.The

The AR(p)model

You want to determine whether or not the unemployment rate for the United States has a stochastic trend using the Augmented Dickey Fuller Test (ADF).The BIC suggests using 3 lags,while the AIC suggests 4 lags. (a)Which of the two will you use for your choice of the optimal lag length? (b)After estimating the appropriate equation,the t-statistic on the lag level unemployment rate is (-2.186)(using a constant,but not a trend).What is your decision regarding the stochastic trend of the unemployment rate series in the United States? (c)Having worked in the previous exercise with the unemployment rate level,you repeat the exercise using the difference in United States unemployment rates.Write down the appropriate equation to conduct the Augmented Dickey-Fuller test here.The t-statistic on relevant coefficient turns out to be (-4.791).What is your conclusion now?

Negative autocorrelation in the change of a variable implies that

You have collected quarterly data on Canadian unemployment (UrateC)and inflation (InfC)from 1962 to 1999 with the aim to forecast Canadian inflation. (a)To get a better feel for the data,you first inspect the plots for the series. Inspecting the Canadian inflation rate plot and having calculated the first autocorrelation to be 0.79 for the sample period,do you suspect that the Canadian inflation rate has a stochastic trend? What more formal methods do you have available to test for a unit root? (b)You run the following regression,where the numbers in parenthesis are homoskedasticity-only standard errors: = 0.49- 0.10 Inft-1 - 0.39 △InfCt-1 - 0.33 △InfCt-2 - 0.21 △InfCt-3 + 0.05 △InfCt-4 (0.28)(0.05)(0.09)(0.09)(0.09)(0.08) Test for the presence of a stochastic trend.Should you have used heteroskedasticity-robust standard errors? Does the fact that you use quarterly data suggest including four lags in the above regression,or how should you determine the number of lags? (c)To forecast the Canadian inflation rate for 2000:I,you estimate an AR(1),AR(4),and an ADL(4,1)model for the sample period 1962:I to 1999:IV.The results are as follows: = 0.002 - 0.31 △InfCt-1 (0.014)(0.10) = 0.021 - 0.46 ΔInfCt-1 - 0.39 ΔInfCt-2 - 0.25 ΔInfCt-3 + 0.03 ΔInfCt-4 (0.158)(0.10)(0.11)(0.08)(0.07) = 1.279 - 0.51 ΔInfCt-1 - 0.44 ΔInfCt-2 - 0.30 ΔInfCt-3 - 0.02 ΔInfCt-4 (0.57)(0.10)(0.11)(0.09)(0.08) - 0.16 UrateCt-1 (0.07) In addition,you have the following information on inflation in Canada during the four quarters of 1999 and the first quarter of 2000: Inflation and Unemployment in Canada,First Quarter 1999 to First Quarter 2000 For each of the three models,calculate the predicted inflation rate for the period 2000:I and the forecast error. (d)Perform a test on whether or not Canadian unemployment rates Granger-cause the Canadian inflation rate.

Pseudo out of sample forecasting can be used for the following reasons with the exception of

Statistical inference was a concept that was not too difficult to understand when using cross-sectional data.For example,it is obvious that a population mean is not the same as a sample mean (take weight of students at your college/university as an example).With a bit of thought,it also became clear that the sample mean had a distribution.This meant that there was uncertainty regarding the population mean given the sample information,and that you had to consider confidence intervals when making statements about the population mean.The same concept carried over into the two-dimensional analysis of a simple regression: knowing the height-weight relationship for a sample of students,for example,allowed you to make statements about the population height-weight relationship.In other words,it was easy to understand the relationship between a sample and a population in cross-sections.But what about time-series? Why should you be allowed to make statistical inference about some population,given a sample at hand (using quarterly data from 1962-2010,for example)? Write an essay explaining the relationship between a sample and a population when using time series.

You have decided to use the Dickey Fuller (DF)test on the United States aggregate unemployment rate (sample period 1962:I - 1995:IV).As a result,you estimate the following AR(1)model t = 0.114 - 0.024 UrateUSt-1,R2 = 0.0118,SER = 0.3417 (0.121)(0.019) You recall that your textbook mentioned that this form of the AR(1)is convenient because it allows for you to test for the presence of a unit root by using the t- statistic of the slope.Being adventurous,you decide to estimate the original form of the AR(1)instead,which results in the following output t = 0.114 - 0.976 UrateUSt-1,R2 = 0.9510,SER = 0.3417 (0.121)(0.019) You are surprised to find the constant,the standard errors of the two coefficients,and the SER unchanged,while the regression R2 increased substantially.Explain this increase in the regression R2.Why should you have been able to predict the change in the slope coefficient and the constancy of the standard errors of the two coefficients and the SER?

The Akaike Information Criterion (AIC)is given by the following formula

(Requires Appendix material)Define the difference operator Δ = (1 - L)where L is the lag operator,such that LjYt = Yt-j.In general, = (1- Lj)i,where i and j are typically omitted when they take the value of 1.Show the expressions in Y only when applying the difference operator to the following expressions,and give the resulting expression an economic interpretation,assuming that you are working with quarterly data: (a)Δ4Yt (b) Yt (c) Yt (d) Yt

Consider the following model Yt = α0 + α1 + ut where the superscript "e" indicates expected values.This may represent an example where consumption depended on expected,or "permanent," income.Furthermore,let expected income be formed as follows: = + λ(Xt-1 - );0 < λ < 1 This particular type of expectation formation is called the "adaptive expectations hypothesis." (a)In the above expectation formation hypothesis,expectations are formed at the beginning of the period,say the 1st of January if you had annual data.Give an intuitive explanation for this process. (b)Transform the adaptive expectation hypothesis in such a way that the right hand side of the equation only contains observable variables,i.e. ,no expectations. (c)Show that by substituting the resulting equation from the previous question into the original equation,you get an ADL(0,∞)type equation.How are the coefficients of the regressors related to each other? (d)Can you think of a transformation of the ADL(0,∞)equation into an ADL(1,1)type equation,if you allowed the error term to be (ut - λut-1)?

The first difference of the logarithm of Yt equals

Consider the AR(1)model Yt = β0 + β1Yt-1 + ut, < 1.. (a)Find the mean and variance of Yt. (b)Find the first two autocovariances of Yt. (c)Find the first two autocorrelations of Yt.

To choose the number of lags in either an autoregression or in a time series regression model with multiple predictors,you can use any of the following test statistics with the exception of the

The ADL(p,q)model is represented by the following equation

The Augmented Dickey Fuller (ADF)t-statistic

Autoregressive distributed lag models include